Was that change real? Quantifying uncertainty for change points

检测数据的变化是目前统计学中最活跃的领域之一。在许多应用中，人们有兴趣将数据分割成具有相同统计特性的区域，作为灵活建模数据的一种方式，以帮助下游分析或确保仅基于相关数据进行预测。而在其他领域，主要兴趣在于检测何时发生变化，因为它们表明了感兴趣的特征，从潜在的机器故障到安全漏洞或基因组特征（例如拷贝数变异）的存在。到目前为止，这一领域的大多数研究都在开发检测变化的方法：输入数据并输出最佳猜测的算法，以确定是否有相关的变化，如果有，有多少变化以及何时发生。一个相对被忽视的问题是评估我们对数据的给定部分发生了特定变化的信心。在许多应用中，量化是否发生了变化的不确定性是至关重要的。例如，如果我们正在监控一个大型通信网络，并且变化表明潜在的故障，那么了解我们对网络中任何给定点存在故障的信心是有帮助的，以便我们可以优先使用有限的资源来调查和修复故障。当分析神经元活动的钙成像数据时，其中变化对应于神经元放电的时间，了解我们在每个时间点有多确定神经元放电是有帮助的，以便改进数据的下游分析。解决这个问题的一种简单方法是首先检测变化，然后应用标准统计测试来确定它们的存在。但这种方法是有缺陷的，因为它使用数据两次，首先决定在哪里测试，然后执行测试。我们可以使用样本分割的想法来克服这个问题-我们使用一半的数据来检测变化，另一半来执行测试。但是这样的方法会失去力量，例如，只使用部分数据来检测变化。该提案将开发统计上有效的方法来量化不确定性，这比样本分割方法更强大。这些方法基于两个互补的想法：（i）在检测之前进行推理;以及（ii）为解释早期检测步骤的变化开发测试。产出将是一个新的通用变化点工具箱，其中包括新的通用统计方法及其在软件包中的实施。

Detecting changes in data is currently one of the most active areas of statistics. In many applications there is interest in segmenting the data into regions with the same statistical properties, either as a way to flexibly model data, to help with down-stream analysis or to ensure predictions are made based only on relevant data. Whilst in others the main interest lies in detecting when changes have occurred as they indicate features of interest, from potential failures of machinery to security breaches or the presence of genomic features such as copy number variations. To date most research in this area has been developing methods for detecting changes: algorithms that input data and output a best guess as to whether there have been relevant changes, and if so how many there have been and when they occurred. A comparatively ignored problem is assessing how confident we are that a specific change has occurred in a given part of the data. In many applications, quantifying the uncertainty around whether a change has occurred is of paramount importance. For example, if we are monitoring a large communication network, and changes indicate potential faults, it is helpful to know how confident we are that there is a fault at any given point in the network so that we can prioritise the use of limited resources available for investigating and repairing faults. When analysing calcium imaging data on neuronal activity, where changes correspond to times at which a neuron fires, it is helpful to know how certain we are that a neuron fired at each time point so as to improve down-stream analysis of the data.A naive approach to this problem is to first detect changes and then apply standard statistical tests for their presence. But this approach is flawed as it uses the data twice, first to decide where to test and then to perform the test. We can overcome this using sample splitting ideas - where we use half the data to detect a change, and the other half to perform the test. But such methods lose power, e.g. from using only part of the data to detect changes. This proposal will develop statistically valid approaches to quantifying uncertainty, that are more powerful than sample splitting approaches. These approaches are based on two complementary ideas (i) performing inference prior to detection; and (ii) develop tests for a change that account for earlier detection steps. The output will be a new general toolbox for change points encompassing both new general statistical methods and their implementation within software packages.